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Trevor Kemchington asks, reasonably enough, to be convinced that as a
rhumb-line approaches the pole, its length remains finite, even though it
makes an infinite number of shrinking turns around the pole as it does so.

Responses from other readers are relevant here. Walter Guinon said "Sounds
a lot like Zeno's paradox to me."
And Bill Noyce said-
>Although a loxodrome spirals around the pole an infinite number of
>times, it has only a finite length.  That's one of the funny things
>about it.  To see this intuitively, imagine it has gotten close enough
>to the pole that the earth can be considered to be flat.  In this
>case, the loxodrome degenerates to a planar logarithmic spiral --
>a spiral that cuts each radius at the same angle.  The logarithmic
>spiral has the property that there is a constant factor f between
>the length of one 360-degree segment and the next.  Therefore, the
>total length of the inward spiral, starting from where the path is
>one mile long, is 1 + f + f^2 + f^3 + ..., which is finite as long
>as f<1.  (The familiar case where f=0.5 adds up to 2.)

Trevor doesn't want too much mathematics and I will try to show him using
simple arithmetic. He says-

>Above a latitude of, say, 89 degrees 50
>minutes, the sphericity of the Earth ought to be negligible and we
>should be able to visualize the problem as one of a rhumb line
>spiralling around the pole in a simple 2-dimensional space, while
>cutting each meridian at the same angle. I can visualize that spiral
>closing in from 10 miles out to a mile and so to a tenth of a mile. But
>it then seems to me that we could simply expand the scale by 100 times
>and see the loxodrome spiralling in from 1/10 of a mile to 1/1000,
>before expanding the scale again and repeating. What I don't see is how
>that loxodrome will eventually make the final step of reducing its
>distance from the pole to zero.

I will take Trevor's word-picture, which describes the situation near the
pole well, presuming that every turn of 360 degrees around the pole reduces
the distance from the pole by a factor of 10. This will apply if a certain
rhumb-line course is chosen to suit. I doubt my own ability to work out
what that course has to be, but for our argument, it doesn't matter. We
presume that the rhumb-line course makes whatever constant angle to true
North is needed to cause a factor-of-10 reduction in the distance from the
pole, each turn.

Not only the radius reduces by a factor of 10, each turn. The distance
travelled along the decreasing spiral in each 360-degree turn around the
pole also reduces by a factor of 10. Trevor clearly understands and accepts
this, in agreeing that after one turn one could remap the picture by
changing the scale by a factor of 10, and then then the next turn would
repeat the same path (but with the important proviso that in real life,
without that scale change, the path would be a factor of 10 shorter).

Consider, as we spiral in toward the pole along this rhumb-line, the
reducing path-length followed in each 360 degree excursion round the pole.
As this shrinks, smoothly, there must be a point P at which the path-length
for the next 360-degree turn will become exactly 1 mile. P will be at a
certain radius from the pole, somewhat less than a mile. What that radius
has to be, I'm not sure how to calculate for such a spiral, but for our
argument that doesn't matter either. It's only important that we know that
such a point P must exist. From that point, we start totting-up the total
path length that we must travel to reach the pole.

In the first 360-degree turn around the pole from P, we agree that the path
length was exactly 1 mile. For the next turn, with everything shrunk by a
factor of 10, it must be 0.1 mile. And for the next, 0.01 mile. And so on,
in ever-decreasing circles, until the vessel's bowsprit disappears up its
own stern-tube...

So the total distance from P, after 1, 2, 3, 4, turns is 1 mile, then 1.1
miles, 1.11 miles, 1.111 miles, and so on, each turn adding a diminishing
amount. After an infinite nunber of turns, the total distance from P will
be 1.11111111... recurring, which is EXACTLY one mile and a ninth. Each
successive turn has brought us closer to that limit, but only after an
infinite number can we claim to have reached one-and-a-ninth miles from P.
Does this answer Trevor's question?

It's an example of a geometric progression, in which each term is a
constant fraction of the previous one. As long as the successive terms are
getting smaller, rather than getting larger, there's always a finite limit
to the sum of the terms.

Herbert Prinz has hypothesised that the speed of the vessel following such
a rhumb-line course is constant, and as a result, because the total
path-length is finite, so must be the time taken.

Because each turn around the pole has a path length only one tenth of the
previous turn, the time taken will be only one-tenth of the previous time.
The vessel must be turning at a faster and faster rate (with respect to the
Universe), to remain on its rhumb-line. Such a theoretical rhumb-line path
could be followed only by a point, a mathematical abstraction, and not by a
real vessel. As such a vessel arrived at the pole, her direction would need
to be changing infinitely fast. She would need to be spinning about the
axis of her mast, requiring infinite rotational energy; an absurd notion.

==========================

Trevor goes on to say-

>While I am willing to accept that I may be wrong and a loxodrome may
>actually reach the pole, it will take a whole lot more to persuade me
>that said loxodrome ever departs from the pole again. George: I really
>think you are wrong on that point.
>
>I _know_ you are wrong on the claim that our hypothetical ship will ever
>leave the pole. You have assumed that the vessel is following a
>rhumb-line course between 270 and 090. Any course away from the North
>Pole must be 180 instantaneously and, immediately thereafter, must be
>between 090 and 270. That is: the hypothetical ship can only leave the
>pole by changing its course (perhaps to its reciprocal) and that is
>contrary to the starting assumption.

On these point, I accept that Trevor is entirely correct, and I was
entirely wrong.

He is also correct to point out that a vessel departing from any latitude
(not just the Equator) with a rhumb-line course of 0 degrees will never
reach eaither pole.


And he is also correct to point out my error in the following statement-

>> At 0deg, the vessel approaches the pole along a certain line of
>> longitude, then emerges along a line of longitude 180deg different.

when he says-

>Again, the vessel cannot leave the North Pole while its course remains
>the rhumb line of 000. As I wrote last time, it can only leave by
>changing to a reciprocal course -- which it would of course do if it
>continued "straight" (meaning on a Great Circle, since this is spherical
>geometry) through the pole. We are, however, assuming rhumb-line
>courses, not reversible rhumb-line courses.

Thanks for pointing out those errors, Trevor.

George.

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